So how would we trace through a real thick lens using our graphical techniques,

or later on our more sophisticated numerical techniques in the paraxial

limit using our thin paraxial techniques, through this thick lens?

So here's how to think about that and we get a very, very simple rule,

but let's not just take it as a rule, let's figure it out.

I'm using the same two rays I had before, 1 and 2, but

I'm going to image turning number 2 around backwards.

So, it's exactly the same ray, but of course, light goes both ways, reciprocity.

So I'm going to imagine now rays 1 and

2 from before are both coming forward at this lens.

Ray 1 bends at the first surface, bends at the second surface,

hits the axis at the back focal point.

Ray 2 comes from the front focal point, bends at the front surface,

bends at the second surface and comes out as an axial ray.

Now, let's think about this in terms of virtual objects and images.

I see rays 1 and 2 here apparently converging to a point right here.

So this is my virtual object.

I have rays 1 and 2 converging to a virtual object which

appears to be right at the front principal point.

Conversely, if I look at whether 1 and 2 is coming out of my lens,

they appear to come from a virtual image that's at the rear principal plane.

So this suggests something kind of interesting.

If I wanted to, for example, let's trace ray 2 through this system.

But I'm trying to use not the real curved surfaces, but just principal planes and

focal lengths, this equivalent, that thick paraxial thin lens system.

I would apparently take ray 2 in, remember, I don't have any glass in

my paraxial ray trace so it just continues to the front principal plane.

It then teleports over to the rear principal plane where I then

apply all of the lens power, capital, or whatever the focal length.

So I bend this ray 2 at this plane,

of course it's coming from the front focal plane so it comes out axially.

Similarly, ray 1 comes in, it goes through this point because I don't know this

surface exists anymore, I've replaced it with it's equivalent description.

It goes right to the point P, teleports to P prime, and

then bends at P prime, of course, because it comes in axially,

goes down through the rear focal point.

The rule is that then you take your curved surfaces of glass, you do

this construction to find the principal planes or for a lens from a catalog,

you just take the principal planes that the catalog manufacturer gives to you.

And if you would like to retrace a system with that real thick lens in it,

you replace no matter how complex it is,

we're sowing singlets that it could be a multi-elemental system.

You replace that whole system with two principal planes and a focal length.

Throw away the real glass now, this is how you go back to

paraxial thin line tracing from a thick optic.

And you shoot rays in following your thin lens conventions.

When you get to the front principal plane,

you teleport those rays with their positions and ray angles intact.

And then at the back principle plane, you apply the length vocal length there,

apply the lens power, as if the thin lens lived there.

Of course if you happen to be going backwards, the same convention applies,

except you teleport from P prime, right to P.

So formally, the principle planes are conjugates and

they know they're conjugates because an object here is

apparently in focus since I've got an image here.

So that's our definition of conjugates and their unity plus one magnification.

So the formal phrase for the principal planes, another more sophisticated way to

think of them is they're the conjugates with unit magnification.

Ray tracing, you simply replace the lens with P and P prime, and

you simply teleport between this gap.

Now warning, sometimes P prime can be over to the left of P.

These planes can be anywhere for unusual negative lens systems, etc.

But it doesn't matter, you just follow the rule.

You shoot your rays in until you hit P, you teleport to P prime,

you'll apply the lens as if it lived on P prime.